Contributions of S-, P-, and D-wave resonances to the quasi-two-body decays $$B^{0}_{s} \rightarrow \psi (3686,3770)K \pi $$ in the perturbative QCD approach

Abstract

AbstractBased on the perturbative quantum chromodynamics (pQCD) approach and the quasi-two-body approximation, we have studied the three-body decays $$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$ B s 0 → ψ ( 3686 , 3770 ) K π , which include the contributions of the intermediate resonances $$\overline{K }^{*}_{0}(1430)^{0}$$ K ¯ 0 ∗ ( 1430 ) 0 , $$\overline{K }^{*}(892)^{0}$$ K ¯ ∗ ( 892 ) 0 , $$\overline{K }^{*}(1410)^{0}$$ K ¯ ∗ ( 1410 ) 0 , $$\overline{K }^{*}(1680)^{0}$$ K ¯ ∗ ( 1680 ) 0 , and $$\overline{K }^{*}_{2}(1430)^{0}$$ K ¯ 2 ∗ ( 1430 ) 0 . The time-like form factors corresponding to the distribution amplitudes of the S-, P-, and D-wave of the kaon–pion pair were adopted in parameterized form, and describe the interactions between $$K $$ K and $$\pi $$ π in the resonance region. First, the decays $$B^{0}_{s}\rightarrow \psi (2S,1D)K ^{-}\pi ^{+}$$ B s 0 → ψ ( 2 S , 1 D ) K - π + were calculated, followed by the calculation of the branching ratios of the decays $$B^{0}_{s}\rightarrow \psi (3686,3770)K ^{-}\pi ^{+}$$ B s 0 → ψ ( 3686 , 3770 ) K - π + using the 2S–1D mixing scheme. In addition, the pQCD predictions for the decays $$B^{0}_{s}\rightarrow \psi (2S,1D)K \pi $$ B s 0 → ψ ( 2 S , 1 D ) K π and $$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$ B s 0 → ψ ( 3686 , 3770 ) K π were obtained using the narrow-width approximation relation given by the Clebsch–Gordan coefficients. Our work shows that the $$\overline{K }^{*}(892)^{0}$$ K ¯ ∗ ( 892 ) 0 resonance is the main contributor to the total decay, and the branching ratio and the longitudinal polarization fraction of the $$\psi (2S)\overline{K }^{*}(892)^{0}$$ ψ ( 2 S ) K ¯ ∗ ( 892 ) 0 decay mode agree well with the currently available data within errors. Furthermore, the theoretical predictions of the $$\psi (2S)$$ ψ ( 2 S ) and $$\psi (3686)$$ ψ ( 3686 ) decay modes are very close, indicating that they can be regarded as the same meson state. Finally, the pQCD predictions for branching ratios of decays $$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$ B s 0 → ψ ( 3686 , 3770 ) K π are of the order of $$10^{-5}$$ 10 - 5 and $$10^{-6}$$ 10 - 6 , respectively, which can be verified using the ongoing LHCb and Belle II experiments.